High Quality Content by WIKIPEDIA articles! A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of an octahedron. The group of orientation-preserving symmetries is S4, or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite sides of the octahedron.

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